Ukrainian soldiers were engaged in fierce fighting on Monday in their country’s northeast, trying to fend off an advance by Russian forces who surged across the border last week to open a new line of attack near the city of Kharkiv.

Russian airstrikes on Monday were pounding Vovchansk, a small town about five miles from the border, according to Denys Yaroslavsky, a Ukrainian officer currently fighting there.

“They’re dropping five to seven bombs every three minutes,” Mr. Yaroslavsky said in a phone interview on Monday, referring to the Russian bombardment.

Vovchansk had a prewar population of about 17,000 people, and local officials have been scrambling to evacuate the estimated 200 to 300 remaining residents. Hryhoriy Shcherban, a volunteer who was in Vovchansk on Monday morning, said that he had received more than 200 requests for evacuation overnight.

“We are driving around trying to find the addresses. Russia is shelling the evacuation road,” he said. “You can hear explosions all the time.”